1. An absent-minded professor has five keys in his pocket, only one of which will open his front door. He randomly takes a key from his pocket, tries it, and if it fails puts it back in his pocket. (a) What is the probability that after four tries the door is still locked? (b) The professor's wife suggests that it might be smarter to lay aside the keys that haven't fit so far. What would be the probability in 1a if the professor had followed his wife's strategy? 2. In a certain country the probability that a newborn survives to age 40 is 0.8 for boys and 0.9 for girls. The corresponding probabilities for surviving to age 80 are 0.2 and 0.3, respectively. Alice and Bob1 are both born on the same day. Questions 2a-2c refer to probabilities at the time of birth. Questions 2d-2f are looking for conditional probabilities 40 years later. Assume that Bob's death at any age is independent of any event involving the survival of Alice. (a) What is the probability that Bob dies before his 40th birthday? (b) What is the probability that Alice will die between the ages of 40 and 80? (c) What is the probability that Bob and Alice are both still alive at their 40th birthday.? (d) Given that Alice sees her 40th birthday, what is the conditional probability that she will see her 80th birthday? (e) Bob and Alice win the lottery at age 40, marry2, and draw up a will which leaves everything to the surviving spouse. On the death of the surviving spouse everything goes to the University. What is the (conditional) probability that the University will get their fortune within the next 40 years? (f) What is the probability that Alice will be a widow at age 80? 3. A factory producing shoddy refrigerators runs three shifts. Two day shifts produce 40% of the output each, whereas the third, night shift produces 20%. Quality control has determined that 10% of the first shift's production is defective, while the second and third shifts have defective rates of 7% and 5% respectively. Interpret these values as empirical probabilities regarding the shift and quality of a randomly selected refrigerator. (a) What is the probability that a randomly selected refrigerator from the factory's overall pro- duction will be defective? (b) A randomly selected refrigerator is found to be defective. What is the probability that it came from the first shift?

1Alice is female and Bob is male.2This makes the assumption of independence suspect, since the death of one spouse can affect the death of another, either directly, such as by a joint accident, or by reducing the will to live in the survivor. However, to keep things simple we will continue assume that such dependence does not occur.